Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93814 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7080 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| prime radical | C23 | #359 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,13,9,46,38,101,10,17,39,61,8,16,37,60)(3,18,20,68,24,84,79,124,25,71,80,121,23,70,78,120)(4,26,29,85,33,96,94,128,34,56,35,97,32,87,93,126)(6,40,42,102,43,14,62,64,7,44,36,99,28,15,65,59)(11,47,48,74,67,105,118,19,52,72,108,75,51,104,107,76)(12,53,55,95,86,113,127,31,58,92,116,91,57,112,115,30)(21,81,82,125,41,69,103,122,22,83,77,123,45,54,100,114)(27,63,88,119,90,49,109,110,73,66,98,117,89,50,111,106);(1,3,10,25)(2,11,17,52)(4,27,34,73)(5,20,39,80)(6,41,7,45)(8,24,9,23)(12,54,58,69)(13,48,61,108)(14,63,15,66)(16,67,46,51)(18,26,71,56)(19,30,76,31)(21,28,22,43)(29,88,35,98)(32,90,33,89)(36,100,42,103)(37,79,38,78)(40,49,44,50)(47,92,72,53)(55,114,116,122)(57,81,86,83)(59,117,64,119)(60,118,101,107)(62,82,65,77)(68,85,121,97)(70,96,84,87)(74,95,75,91)(93,109,94,111)(99,106,102,110)(104,112,105,113)(115,125,127,123)(120,128,124,126)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1013);